 # Intermediate Algebra: Functions and Graphs

## Section9.6Chapter 9 Summary and Review

### Subsection9.6.1Glossary

• horizontal
• vertical
• parallel
• perpendicular
• conic section
• circle
• ellipse
• parabola
• hyperbola
• central conic
• major axis
• minor axis
• vertices
• covertices
• transverse axis
• conjugate axis
• asymptote
• branch

### Subsection9.6.2Key Concepts

1. #### Horizontal and Vertical Lines.

1. The equation of the horizontal line passing through $$(0, b)$$ is
\begin{gather*} y=b \end{gather*}
2. The equation of the vertical line passing through $$(a,0)$$ is
\begin{gather*} x=a \end{gather*}
2. #### Slopes of Horizontal and Vertical Lines.

The slope of a horizontal line is zero.
The slope of a vertical line is undefined.
3. #### Parallel and Perpendicular Lines.

1. Two lines are parallel if their slopes are equal, that is, if
\begin{gather*} m_1 = m_2 \end{gather*}
or if both lines are vertical.
2. Two lines are perpendicular if the product of their slopes is $$-1\text{,}$$ that is, if
\begin{gather*} m_1m_2 = -1 \end{gather*}
or if one of the lines is horizontal and one is vertical.
4. The distance between two points is the length of the segment joining them.
5. #### Distance Formula.

The distance $$d$$ between points $$P_1(x_1, y_1)$$ and $$P_2(x_2, y_2)$$ is
\begin{gather*} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \end{gather*}
6. The midpoint of a segment is the point halfway between its endpoints, so that the distance from the midpoint to either endpoint is the same.
7. #### Midpoint Formula.

The midpoint of the line segment joining the points $$P_1(x_1, y_1)$$ and $$P_2(x_2, y_2)$$ is the point $$M(\overline{x},\overline{y})$$ where
\begin{gather*} \overline{x}=\frac{x_1+x_2}{2} \quad\text{ and } \overline{y}=\frac{y_1+y_2}{2} \end{gather*}
8. A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center.
9. #### Circle.

The equation for a circle of radius $$r$$ centered at the point $$(h, k)$$ is
\begin{gather*} (x-h)^2+(y-k)^2=r^2 \end{gather*}
10. #### Ellipse.

The standard form for an ellipse centered at $$(h, k)$$ is
\begin{gather*} \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \end{gather*}
11. #### Hyperbola.

The equation for a hyperbola centered at the point $$(h, k)$$ has one of the two standard forms:
\begin{gather*} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \end{gather*}
\begin{gather*} \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 \end{gather*}
12. #### Conic Sections.

The graph of $$Ax^2+Cy^2+Dx+Ey+F=0$$ is
1. a circle of $$A=C\text{.}$$
2. a parabola if $$A=0$$ or $$C=0$$ (but not both).
3. an ellipse if $$A$$ and $$C$$ have the same sign.
4. a hyperbola if $$A$$ and $$C$$ have opposite signs.
13. A system of two quadratic equations may have up to four solutions.

### Exercises9.6.3Chapter 9 Review Problems

#### Exercise Group.

For Problems 1 and 2, decide whether the lines are parallel, perpendicular, or neither.
##### 1.
$$y=\dfrac{1}{2}x+3 \text{;}$$ $$\quad x-2y=8$$
parallel
##### 2.
$$4x-y=6 \text{;}$$ $$\quad x+4y=-2$$
perpendicular

#### 3.

Write an equation for the line that is parallel to the graph of $$2x + 3y = 6$$ and passes through the point $$(1, 4)$$
$$y=\dfrac{-2}{3}x+\dfrac{14}{3}$$

#### 4.

Write an equation for the line that is perpendicular to the graph of $$2x + 3y = 6$$ and passes through the point $$(1, 4)$$
$$y=\dfrac{3}{2}x+\dfrac{5}{2}$$

#### 5.

Two vertices of the rectangle $$ABCD$$ are $$A(3, 2)$$ and $$B(7, −4)\text{.}$$ Find an equation for the line that includes side $$\overline{BC}\text{.}$$
$$y=\dfrac{2}{3}x-\dfrac{26}{3}$$

#### 6.

One leg of the right triangle $$PQR$$ has vertices $$P(-8, -1)$$ and $$Q(-2, -5)\text{.}$$ Find an equation for the line that includes the leg $$\overline{QR} \text{.}$$
$$y=\dfrac{3}{2}x-2$$

#### 7.

Find the perimeter of the triangle with vertices $$A(-1, 2)\text{,}$$ $$B(5, 4)\text{,}$$ $$C(1, -4)\text{.}$$ Is $$\Delta ABC$$ a right triangle?
21.59; yes

#### 8.

Find the midpoint of each side of $$\Delta ABC$$ from the previous problem. Join the midpoints to form a new triangle, and find its perimeter.
10.8

#### Exercise Group.

For Problems 9–18, graph the conic section.
##### 9.
$$x^2+y^2=9$$ ##### 10.
$$\dfrac{x^2}{9}+y^2=1$$ ##### 11.
$$\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$$ ##### 12.
$$\dfrac{y^2}{6}-\dfrac{x^2}{8}=1$$ ##### 13.
$$4(y-2)=(x+3)^2$$ ##### 14.
$$(x-2)^2+(y+3)^2=16$$ ##### 15.
$$\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1$$ ##### 16.
$$\dfrac{(x+4)^2}{12}+\dfrac{(y-2)^2}{6}=1$$ ##### 17.
$$\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1$$ ##### 18.
$$(x-2)^2+4y=4$$ #### Exercise Group.

For Problems 19–30
1. Write the equation of each conic section in standard form.
2. Identify the conic and describe its main features.
##### 19.
$$x^2+y^2-4x+2y-4=0$$
1. $$\displaystyle (x-2)^2+(y+1)^2=9$$
2. Circle: center $$(2,-1)\text{,}$$ radius 3
##### 20.
$$x^2+y^2-6y-4=0$$
1. $$\displaystyle x^2+(y-3)^2=13$$
2. Circle: center $$(0,3)\text{,}$$ radius $$\sqrt{13}$$
##### 21.
$$4x^2+y^2-16x+4y+4=0$$
1. $$\displaystyle \dfrac{(x-2)^2}{4}+\dfrac{(y+2)^2}{16}=1$$
2. Ellipse: center $$(2,-2)\text{,}$$ $$a=2\text{,}$$ $$b=4$$
##### 22.
$$8x^2+5y^2+16x-20y-12=0$$
1. $$\displaystyle \dfrac{(x+1)^2}{5}+\dfrac{(y-2)^2}{8}=1$$
2. Ellipse: center $$(-1,2)\text{,}$$ $$a=\sqrt{5}\text{,}$$ $$b=\sqrt{8}$$
##### 23.
$$x^2-8x-y+6=0$$
1. $$\displaystyle y+10=(x-4)^2$$
2. Parabola: vertex $$(4,10)\text{,}$$ opens upward, $$a=1$$
##### 24.
$$y^2+6y+4x+1=0$$
1. $$\displaystyle x-2=\dfrac{-1}{4}(y+3)^2$$
2. Parabola: vertex $$(2,-3)\text{,}$$ opens left, $$a=\dfrac{-1}{4}$$
##### 25.
$$x^2+y=4x-6$$
1. $$\displaystyle y+2=-(x-2)^2$$
2. Parabola: vertex $$(2,-2)\text{,}$$ opens downward, $$a=-1$$
##### 26.
$$y^2=2y+2x+2$$
1. $$\displaystyle x+\dfrac{3}{2}=\dfrac{1}{2}(y-1)^2$$
2. Parabola: vertex $$\left(\dfrac{-3}{2},1\right)\text{,}$$ opens right, $$a=\dfrac{1}{2}$$
##### 27.
$$2y^2-3x^2-16y-12x+8=0$$
1. $$\displaystyle \dfrac{(y-4)^2}{6}-\dfrac{(x+2)^2}{4}=1$$
2. Hyperbola: center $$(-2,4)\text{,}$$ transverse axis vertical, $$a=2\text{,}$$ $$b=\sqrt{6}$$
##### 28.
$$9x^2-4y^2-72x-24y+72=0$$
1. $$\displaystyle \dfrac{(x-4)^2}{4}-\dfrac{(y+3)^2}{9}=1$$
2. Hyperbola: center $$(4,-3)\text{,}$$ transverse axis horizontal, $$a=2\text{,}$$ $$b=3$$
##### 29.
$$2x^2-y^2+6y-19=0$$
1. $$\displaystyle \dfrac{x^2}{5}-\dfrac{(y-3)^2}{10}=1$$
2. Hyperbola: center $$(0,3)\text{,}$$ transverse axis horizontal, $$a=\sqrt{5}\text{,}$$ $$b=\sqrt{10}$$
##### 30.
$$4y^2-x^2+8x-28=0$$
1. $$\displaystyle \dfrac{y^2}{3}-\dfrac{(x-4)^2}{12}=1$$
2. Hyperbola: center $$(4,0)\text{,}$$ transverse axis vertical, $$a=2\sqrt{3}\text{,}$$ $$b=\sqrt{310}$$

#### 31.

An ellipse centered at the origin has a vertical major axis of length 16 and a horizontal minor axis of length 10.
1. Find the equation of the ellipse.
2. What are the values of $$y$$ when $$x = 4\text{?}$$
1. $$\displaystyle \dfrac{x^2}{25}+\dfrac{y^2}{64}=1$$
2. $$\displaystyle \dfrac{\pm 24}{5}$$

#### 32.

An ellipse centered at the origin has a horizontal major axis of length 26 and a vertical minor axis of length 18.
1. Find the equation of the ellipse.
2. What are the values of $$y$$ when $$x = 12\text{?}$$
1. $$\displaystyle \dfrac{x^2}{169}+\dfrac{y^2}{81}=1$$
2. $$\displaystyle \dfrac{\pm 45}{13}$$

#### Exercise Group.

For Problems 33–38, write an equation for the conic section with the given properties.
##### 33.
Circle: center at $$(-4, 3)\text{,}$$ radius $$2\sqrt{5}$$
$$(x+4)^2+(y-3)^2=20$$
##### 34.
Circle: endpoints of a diameter at $$(-5, 2)$$ and $$(1, 6)$$
$$(x+2)^2+(y-4)^2=13$$
##### 35.
Ellipse: center at $$(-1, 4)\text{,}$$ $$a = 4\text{,}$$ $$b = 2$$
$$\dfrac{(x+1)^2}{16}+\dfrac{(y-4)^2}{4}=1$$
##### 36.
Ellipse: vertices at $$(3, 6)$$ and $$(3, -4)\text{,}$$ covertices at $$(1, 1)$$ and $$(5, 1)$$
$$\dfrac{(x-3)^2}{4}+\dfrac{(y-1)^2}{25}=1$$
##### 37.
Hyperbola: center $$(2, -3)\text{,}$$ $$a = 4\text{,}$$ $$b = 3\text{,}$$ transverse axis horizontal
$$\dfrac{(x-2)^2}{16}-\dfrac{(y+3)^2}{9}=1$$
##### 38.
Hyperbola: one vertex at $$(−4, 1)\text{,}$$ one end of the vertical conjugate axis at $$(-3, 4)$$
$$(x+3)^2 -\dfrac{(y-1)^2}{9}=1$$

#### Exercise Group.

For Problems 39–42, graph the system of equations and state the solutions of the system
##### 39.
\begin{aligned}[t] 4x^2+y^2=25\\ x^2-y^2=-5 \end{aligned}
$$(\pm 2, \pm 3)$$
##### 40.
\begin{aligned}[t] 4x^2-9y^2+132=0\\ x^2+4y^2-67=0 \end{aligned}
$$(\pm\sqrt{3}, \pm 4)$$
##### 41.
\begin{aligned}[t] x^2+3y^2=13\\ xy=-2 \end{aligned}
$$(1,-2) \text{,}$$ $$(-1,2) \text{,}$$ $$\left(2\sqrt{3}, \dfrac{-1}{\sqrt{3}} \right) \text{,}$$ $$\left(-2\sqrt{3}, \dfrac{1}{\sqrt{3}} \right)$$
##### 42.
\begin{aligned}[t] x^2+y^2=17\\ 2xy=-17 \end{aligned}
$$\left(\dfrac{\sqrt{34}}{2}, \dfrac{\sqrt{34}}{2} \right) \text{,}$$ $$\left(\dfrac{-\sqrt{34}}{2}, \dfrac{-\sqrt{34}}{2} \right)$$

#### Exercise Group.

For Problems 43–48, write and solve an equation or a system of equations
##### 43.
Moia drives 180 miles in the same time that Fran drives 200 miles. Find the speed of each if Fran drives 5 miles per hour faster than Moia.
Moia: 45 mph, Fran: 50 mph
##### 44.
The perimeter of a rectangle is 26 inches and the area is 12 square inches. Find the dimensions of the rectangle.
12 in by 1 in
##### 45.
The perimeter of a rectangle is 34 centimeters and the area is 70 square centimeters. Find the dimensions of the rectangle.
Hattie’s annual income from an investment is $32. If she had invested$200 more and the rate had been 1/2% less, her annual income would have been $35. What are the amount and rate of Hattie’s investment? Answer. Amount:$800, rate: 4%